The generalized Lax conjecture holds for mixed determinants
Source: hexagonmath.org/abs/2611.00117v1
We prove that the mixed determinant $\eta(\cdot)$ of Borcea and Brändén has a spectrahedral hyperbolicity cone. More precisely, for Hermitian matrices $A_1,\ldots,A_m$ of order $d$, the hyperbolicity cone of $\eta(A_1,\ldots,A_m)$ is cut out by $\sum_j\Phi_j(A_j)\succeq0$, where $\Phi_1,\ldots,\Phi_m$ are explicit unital completely positive maps. We argue by induction on $d$, adding one row and one column to every argument and expressing the logarithmic derivatives of $\eta$ as extremal eigenvalues. By compressing along polynomial boundary kernels, we reduce the order of the pencil…
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